Theorem pnrmtop 22844

Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmtop	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop

Step	Hyp	Ref	Expression
1		pnrmnrm 22843	. 2 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2		nrmtop 22839	. 2 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2106  Topctop 22394  Nrmcnrm 22813  PNrmcpnrm 22815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551  df-ov 7411  df-nrm 22820  df-pnrm 22822

This theorem is referenced by:  pnrmopn  22846